L11n300

L11n299

L11n301

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Link Presentations

[edit Notes on L11n300's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X18,13,19,14 X17,11,18,22 X7,17,8,16 X21,8,22,9 X14,10,15,9 X20,16,21,15 X10,19,5,20 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -5, 6, 7, -9}, {11, -2, 3, -7, 8, 5, -4, -3, 9, -8, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{(t(1)-1) (t(2)+t(3)-1) (t(3) t(2)-t(2)-t(3))}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^3-3 q^2+4 q-5+7 q^{-1} -5 q^{-2} +6 q^{-3} -3 q^{-4} +2 q^{-5} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 z^{-2} +a^4 z^2-2 a^4 z^{-2} -2 a^4-a^2 z^4+a^2 z^{-2} +z^2 a^{-2} +2 a^2-z^4-z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^3 z^9+a z^9+a^4 z^8+4 a^2 z^8+3 z^8-3 a^3 z^7+3 z^7 a^{-1} -4 a^4 z^6-16 a^2 z^6+z^6 a^{-2} -11 z^6+a^5 z^5+3 a^3 z^5-9 a z^5-11 z^5 a^{-1} +8 a^4 z^4+21 a^2 z^4-3 z^4 a^{-2} +10 z^4+8 a z^3+8 z^3 a^{-1} +2 a^6 z^2-2 a^4 z^2-9 a^2 z^2+z^2 a^{-2} -4 z^2+2 a^5 z+2 a^3 z-2 a^6-3 a^4-2 a^2-2 a^5 z^{-1} -2 a^3 z^{-1} +a^6 z^{-2} +2 a^4 z^{-2} +a^2 z^{-2} }[/math] (db)

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 χ 7                 1 1 5               2   -2 3             2 1   1 1           3 2     -1 -1         4 2       2 -3       2 4         2 -5     4 3           1 -7   1 4             3 -9 1 2               -1 -11 2                 2

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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